Thin Lenses

Problems from IIT JEE

Problem (IIT JEE 1997): Two thin lenses, when in contact, produce a combination of power $+10\,D$. When they are 0.25 m apart, the power reduces to $+6\,D$. The focal length of the lenses are ______ m and _______ m.

Solution: The effective focal length of two thin lenses of focal length $f_1$ and $f_2$ in contact is, \begin{align} \label{qvb:eqn:1} {1}/{F}={1}/{f_1}+{1}/{f_2}, \end{align} and that for lenses separated by a distance $d$ is, \begin{align} \label{qvb:eqn:2} {1}/{F^\prime}={1}/{f_1}+{1}/{f_2}-{d}/{(f_1 f_2)}. \end{align} Substitute ${1}/{F}={10}\;\mathrm{m^{-1}}$ in first equation, ${1}/{F^\prime}={6}\;\mathrm{m^{-1}}$ and $d={0.25}\;\mathrm{m}$ in second equation and then solve to get $f_1={{1}/{8}}\;\mathrm{m}$ and $f_2={{1}/{2}}\;\mathrm{m}$.